Crimp I & II – Peirce Geometry
The jammed plain weave
Peirce modelled a yarn in cloth as a circular cross-section following arcs around the crossing threads. When the threads touch — the jammed state — the yarn has no straight portion; it is all arc. With D = d₁ + d₂ (sum of diameters), the thread spacing p, yarn length l per repeat and crimp height h depend only on the angle θ at which the yarn leaves the crossing thread:
In the drawings the yellow circles are the picks seen end-on and the blue thread is a warp end wrapping them. Edit the diameters below to see every quantity change.
Crimp calculator
| A | B | C | |
|---|---|---|---|
| 1 | Warp (1) | Weft (2) | |
| 2 | Diameter d | ||
| 3 | D = d₁ + d₂ | 2.000 | |
| 4 | Crimp height h (= other yarn’s diameter when jammed) | 1.000 | 1.000 |
| 5 | Angle θ (degrees) | 60.0 | 60.0 |
| 6 | Thread spacing p (warp spacing uses θ₂, weft spacing θ₁) | 1.732 | 1.732 |
| 7 | Yarn length per crossing l = Dθ | 2.094 | 2.094 |
| 8 | Crimp % = l ÷ (spacing of the crossing threads) − 1 | 20.9% | 20.9% |
| 9 | Cover factor K = 2670 × d / p | 1,542 | 1,542 |
Twill and warp rib
2/2 twill, equal yarns
Each thread crosses only every second thread, so one repeat spans 2p = 2.732 d (one jammed arc plus a straight float of length d): K = 2670 × 2 ÷ 2.732 = 1955. Yarn length per repeat l = 3.09 d, crimp = 3.09 ÷ 2.732 − 1 = 13.1%. Twills are less crimped and can be set denser than plain weave — the reason for the weave divisors in Lecture 12.
Warp rib
Warp ends lie side by side (p₁ = d₁, K₁ = 2670) and each passes over two picks. The weft spacing is 1.5 d₂ (K₂ ≈ 1760), the warp wraps a half circle of diameter 3d₁: l₁ = 4.71 d₁ over 3 d₁ of cloth, so warp crimp is about 57% while the straight weft has none. That is why rib fabrics use so much more warp yarn than their weight suggests.
Flattened (non-circular) yarns cover more width than the circular model predicts: real plain cloths look fully covered at K ≈ 1000–1400 rather than 1542.